| In this dissertation, we investigate intuitionistic fuzzy subsets of ordered Γ-seinigroups as follows:intuitionistic fuzzy left (right) ideal, intuitionistic fuzzy bi-ideal, intuitionistic fuzzy quasi-ideal, intuitionistic fuzzy interior-ideal and in-tuitionistic fuzzy normal interior-ideal and get some properties of theirs, by which we characterize the regularity and intra-regularity of ordered Γ-semigroups. There are there chapters, the main results are given as following.In chapter1, we mainly give some basic concepts, symbols and lemmas in this dissertation.In the first section of chapter2, we mainly give the basic concepts of intuition-istic fuzzy left (right) ideal of ordered Γ-semigroups and some properties of theirs. The main results are given as following:Theorem2.1.2Let A=(μA,γA) be an intuitionistic fuzzy right (left) ideal of ordered Γ-semigroups S. Then A U (1～o A)(A U(Aο1～)) is an intuitionistic fuzzy ideal of S.In the second section of chapter2, we mainly give the concepts of intuitionistic fuzzy bi-ideal and intuitionistic fuzzy quasi-ideal of ordered Γ-semigroups and study some properties of theirs. The main results are given as following:Theorem2.2.1Let X be a non-empty subset of ordered Γ-semigroup S. Then X is a bi-ideal of S if an only if χx=(μχx,γχx) is an intuitionistic fuzzy bi-ideal of S.Theorem2.2.2Let{Ai,i∈I} be a collection of intuitionistic fuzzy bi-idcal of ordered Γ-semigroup S. Then∩i∈I Ai is an intuitionistic fuzzy bi-ideal of S.Theorem2.2.3If A=(μA,γA) is an intuitionistic fuzzy bi-ideal of ordered Γ-scmigroup S. Then AοA C A and A o1～ο A (?) A.Theorem2.2.4Let A=(μA,γA) be an intuitionistic fuzzy subset of ordered Γ-semigroup S. If A οA(?)A, Aο1～ο A(?) A and μA(x)≥μA(y),γA(x)<γA(y) for all x.y∈S.x≤y. then A is an intuitionistic fuzzy bi-ideal of ordered Γ-semigroup S.Theorem2.2.5Let A=(μA,γA) be an intuitionistic fuzzy bi-ideal of ordered Γ-semigroup S. Then for any t∈ImμA∩ImγA,U(μA;t) and L(γA:t) are bi-ideals of S.Theorem2.2.6Let A=(μA,γA)be an intuitionistic fuzzy subset of ordered Γ-semigroup S. If for any t∈[0,1],U(μA;t) and L(γA;t) are non-empty sets, and they are bi-idcals of S, then A is an intuitionistic fuzzy bi-ideal of S.Theorem2.2.7Let X be a non-empty subset of ordered Γ-semigroup S. Then X is a quasi-ideal of S if and only if χx=(μχx,γχx) is an intuitionistic fuzzy quasi-ideal of S.Theorem2.2.8Let S be an ordered Γ-semigroup. Then an intuitionistic fuzzy left ideal of S is an intuitionistic fuzzy quasi-ideal of S and an intuitionistic fuzzy right ideal of S is an intuitionistic fuzzy quasi-ideal of S.Theorem2.2.9Let S be an ordered Γ-semigroup, A∈IFΓLI, B∈IFΓRI. Then A∩B∈IFΓQI.Theorem2.2.10Let A=(μA,γA) be an intuitionistic fuzzy quasi-ideal of ordered Γ-scmigroup S. Then A=[AU(1～ο A)]∩[A U (A o1～)].Theorem2.2.11Let A=(μA,γA)be an intuitionistic fuzzy subset of ordered Γ-semigroup S. Then A∈IFΓQI if and only if there exist B∈IFΓLI, C∈IFΓRI such that A=B∩C.Theorem2.2.12Let{Ai,i∈I} be a collection of intuitionistic fuzzy quasi-ideal of ordered Γ-semigroup S. Then∩i∈I Ai is an intuitionistic fuzzy quasi-ideal of S.Theorem.2.2.13An intuitionistic fuzzy quasi-ideal of ordered Γ-semigroup S is an intuitionistic fuzzy bi-ideal of S.Theorem2.2.14Let A be an intuitionistic fuzzy quasi-ideal of ordered Γ-semigroup S and B be an intuitionistic fuzzy subset of S. Then A o B and B o A are intuitionistic fuzzy bi-ideals of S. In the third section of chapter2, we define the intuitionistic fuzzy interior-ideal and intuitionistic fuzzy normal interior-ideal of ordered Γ-semigroups and give some properties of theirs. The main results are given as following:Theorem2.3.1Let X be a non-empty subset of ordered Γ-semigroup S. Then X is a interior-ideal of S if and only if χx=(μχx,γχx) is an intuitionistic fuzzy interior-ideal of S.Theorem2.3.2Let A=(μA,γA) be an intuitionistic fuzzy interior-ideal of ordered Γ-semigroup S. Then1～οAο1～(?)A.Theorem2.3.3Let A=(μA,γA) be an intuitionistic fuzzy subset of ordered Γ-semigroup S. If1～οAο1～(?)A. and μA(x)≥μA(y),γA(X)≤γA(y) for all x,y∈S,x≤y, then A is an intuitionistic fuzzy interior-ideal of S.Theorem2.3.4Let{Ai, i∈I} be a collection of intuitionistic fuzzy interior-ideal of ordered Γ-semigroup S. Then∩∫∈I Ai is an intuitionistic fuzzy interior-ideal of S.Theorem2.3.5Let A=(μA,γA)be an intuitionistic fuzzy interior-ideal of ordered Γ-scmigroup S. The for all t∈ImpμA∩ImγA U(μA;t) and L(γA;t) are interior-ideals of S.Theorem2.3.6Let A=(μA,γA) be an intuitionistic fuzzy subset of ordered Γ-semigroup S. If for all t∈[0.1], U(μA;t) and L(γA,t) are non-empty sets, and they are interior-ideals of S. then A is an intuitionistic fuzzy interior-ideal of S.Theorem2.3.7Let A=(μA,γA) be an intuitionistic fuzzy interior-ideal of ordered Γ-semigroup S. Then SA is an interior-ideal of S.Theorem2.3.8If X is an interior-ideal of S, then χx=(μχx,γχx) is an intuitionistic fuzzy normal interior-ideal of S and SXx=X.Theorem2.3.9If A=(μA,γA) and B=(μB,γB) are intuitionistic fuzzy normal interior-ideals of S, then Sa∩SB=SA∩B.Theorem2.3.10Let A=(μA,γA) and B=(μB,γB) be intuitionistic fuzzy interior-ideals of ordered Γ-semigroup S. If A C B and μA(0)=μB(0),γ4(0)=γB(0):then SA (?)SB.Theorem2.3.11Let A=(μA,γA) be an intuitionistic fuzzy interior-ideal. We have the following definition" Then A+=(μA,γA+)is an intuitionistic fuzzy normal interior-ideal of S and4(?)A+.Theorem2.3.12Let.4=(μA,γA)be an intuitionistic fuzzy interior-ideal. Then A is normal if and only if A+=A.Theorem2.3.13Let A∈IFΓII, If there exists B∈IFΓII such that B+(?)A.Then A is normal.Theorem2.3.14Let A=(μA,γA)be an intuitionistic fuzzy interior-ideal of S. μθ:[0,μA(0)]â†'[0,1]and γθ:[γA(0),1â†'[0,1]are monotone increasing functions. We define the following intuitionistic fuzzy subset of S: Then Aθis an intuitionistic fuzzy interior-ideal of S.If μθ(μA(0))=1,γθ(γA(0))=0. Thcn Aθis normal. In particular,if for any t∈[0,μA(0)],μθ(t)≥t and for all t∈[γA(0),1],γθ(t)≤t,then A(?)AG.Theorem2.3.15Let A=(μA,γA)be an maximal intuitionistic fuzzy normal interior-ideal of S (about the relation of inclusion within the normal intuitionis-tic fuzzy interior-ideals)and it’s not constant complex mapping. Then,ImμA {0,1},ImγA={0,1}.In the one section of chapter3,the regular ordered Γ-semigroups are charac-terized by some kinds of intuitionistic fuzzy ideals. The main results are given as following:Theorem3.1.1Let S be an ordered Γ-semigroup. Then the following are equivalent:(1)S is regularï¼›(2)For each A∈,IFΓRI and each B∈,IFΓLI,AοB=A∩Bï¼›(3)For ea,ch A∈,IFΓRI and each B∈,IFΓLI,(a)AοA=A,(b)BοB=B,(c)AοB∈IFΓQI:(4)(IFΓQI,ο,(?))is a rcgular ordered semigroupï¼›(5)Every A∈IFΓQI has the form A=Aο1～οA. Theorem3.1.2Let S be an regular ordered Γ-semigroup and A=(μ.4.γ4) be an intuitionistic fuzzy subset of S. Then A∈IFΓQI if and only if there exist B∈IFTRI, C∈IFTLI such that A=BoC.Theorem3.1.3Let S be an regular ordered Γ-semigroup. Then A∈IFYQI if and only if A∈IFTBI.Theorem3.1.4Let S be an regular ordered Γ-semigroup. Then the following are equivalent:(1) For each A∈IFTRI and each B∈IFTLI, A o B=An B (?) B o A;(2) Semigroup (IFΓQI, o) is a band;(3) For each A∈IFΓQI, A o A=A.Theorem3.1.5Let S be an ordered Γ-semigroup. Then S is regular if and only if for each A∈IFΔQI and each B∈IFΓI, AoBoA=AnB.Theorem3.1.6Let S be an ordered Γ-semigroup. Then S is regular if and only if A, B∈IFΓLI, C∈IFΓBI, C∩AnB(?)CoAoB.Theorem3.1.7Let S be an ordered Γ-scmigroup. Then S is regular if and only if A,B∈IFΓBI and C∈IFΔI I, A∩CnB (?)AoCoB.In the second section of chapter3, the intra-regular ordered Γ-semigroups are characterized by some kinds of intuitionistic fuzzy ideals. The main results are given as following:Theorem3.2.1ordered Γ-semigroup S is intra-regular if and only if for each A∈IFΓRI and B∈IFΓLI, AnB C Bo A.Theorem3.2.2Let A=(μA,γA) is an intuitionistic fuzzy subscmigroup of ordered Γ-semigroup S. then A is semiprime if and only if for all a∈S exists γ∈Γ such that A(a)=A(a-ya).Theorem3.2.3Let S be an intra-regular ordered Γ-scmigroup and A=(μA,γA) is an intuitionistic fuzzy subset of S. Then A is an intuitionistic fuzzy ideal of S if and only if A is an intuitionistic fuzzy intra-ideal of S.Theorem3.2.4Let S be an ordered Γ-semigroup. Then S is intra-regular if and only if Γ∈IFΓLI, B∈IFΓBI, C∈IFΓRI, A∩B(?)C CoBoC.Theorem3.2.5Let S be an ordered Γ-semigroup. Then S is intra-regular if and only if A∈IFΓLI, B∈IFΓRI, C∈IFΓI I, A∩B∩C(?) AoBoC. |
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